• Uber is a ridesharing service that services millions of customers each day
• Data were collected on individual rides in New York City from September 2014 to August 2015
• Drivers want to know how to make the most money

We are interested in answering the following questions:

• What are the times in the year when the most money can be made?
• What are the times of day when the most money can be made?
• Which locations are the busiest?
• What routes are the most profitable?

\[Y(t) = f(t) + \epsilon(t) \ \text{where} \ \epsilon(t) \sim \mathcal{N}(0, \sigma^2)\]

\[\boldsymbol{Y} | \boldsymbol{f} \sim \mathcal{N}(\boldsymbol{f}, \sigma^2 I)\]

\[\boldsymbol{f} \sim \mathcal{N}(\boldsymbol{0}, K_f) \ \text{where} \ K_f(t, t') = \tau^2 k(t, t')\]

\[\begin{bmatrix} \boldsymbol{Y} \\ \boldsymbol{f} \end{bmatrix} \sim \mathcal{N} \Bigg( \begin{bmatrix} \boldsymbol{0} \\ \boldsymbol{0} \end{bmatrix} , \begin{bmatrix} \sigma^2 I + K_f & K_f \\ K_f & K_f \end{bmatrix}\Bigg)\]

\[\boldsymbol{f} | \boldsymbol{Y} \sim \mathcal{N}(K_f (\sigma^2 I + K_f)^{-1} \boldsymbol{Y}, K_f - K_f(\sigma^2 I + K_f)^{-1}K_f)\]

\[Y(t) = f_{long}(t) + f_{medium}(t) + f_{short}(t) + \epsilon(t)\]

\[f_{*}(t) \sim \mathcal{N}(\boldsymbol{0}, K_*(t,t'))\]

\[K_*(t,t') = \tau^2_* \text{exp}\Bigg(-\frac{1}{2 l^2_*} |t-t'|^2\Bigg)\]

\[Y(s,t) = f(s,t) + \epsilon(s, t)\]

\[\boldsymbol{Y} | \boldsymbol{f} \sim \mathcal{N}(\boldsymbol{f}, \sigma^2 I)\]

\[\boldsymbol{f} \sim \mathcal{N}(\boldsymbol{0}, K_S \otimes K_T)\]

\[K_S(s,s') = \tau^2_S \text{exp}\Bigg(-\frac{1}{2 l^2_S} ||s-s'||^2_2\Bigg)\]

\[K_T(t,t') = \tau^2_T \text{exp}\Bigg(-\frac{1}{2 l^2_S} |t-t'|^2\Bigg)\]

• Issue: Currently fixing length-scale parameters to reasonable values
• Possible Solution: Could estimate them
• Issue: Current covariance functions might not accurately represent underlying processes
• Possible Solution: Could consider other covariance functions
• Issue: High autocorrelation for some variance parameters
• Possible Solution: Try implementation in STAN